25 research outputs found
Everything you always wanted to know about the parameterized complexity of Subgraph Isomorphism (but were afraid to ask)
Given two graphs H and G, the Subgraph Isomorphism problem asks if H is isomorphic to a subgraph of G. While NP-hard in general, algorithms exist for various parameterized versions of the problem. However, the literature contains very little guidance on which combinations of parameters can or cannot be exploited algorithmically. Our goal is to systematically investigate the possible parameterized algorithms that can exist for Subgraph Isomorphism.
We develop a framework involving 10 relevant parameters for each of H and G (such as treewidth, pathwidth, genus, maximum degree, number of vertices, number of components, etc.), and ask if an algorithm with running time f1_(p_1,p_2,...,p_l).n^f_2(p_(l+1),...,p_k) exists, where each of p_1,...,p_k is one of the 10 parameters depending only on H or G. We show that all the questions arising in this framework are answered by a set of 11 maximal positive results (algorithms) and a set of 17 maximal negative results (hardness proofs); some of these results already appear in the literature, while others are new in this paper.
On the algorithmic side, our study reveals for example that an unexpected combination of bounded degree, genus, and feedback vertex set number of G gives rise to a highly nontrivial algorithm for Subgraph Isomorphism. On the hardness side, we present W[1]-hardness proofs under extremely restricted conditions, such as when H is a bounded-degree tree of constant pathwidth and G is a planar graph of bounded pathwidth
A tight lower bound for steiner orientation
In the STEINER ORIENTATION problem, the input is a mixed graph G (it has both directed and undirected edges) and a set of k terminal pairs T. The question is whether we can orient the undirected edges in a way such that there is a directed sât path for each terminal pair (s,t)âT. Arkin and Hassin [DAMâ02] showed that the STEINER ORIENTATION problem is NP-complete. They also gave a polynomial time algorithm for the special case when k=2
.
From the viewpoint of exact algorithms, Cygan, Kortsarz and Nutov [ESAâ12, SIDMAâ13] designed an XP algorithm running in nO(k) time for all kâ„1. Pilipczuk and Wahlström [SODA â16] showed that the STEINER ORIENTATION problem is W[1]-hard parameterized by k. As a byproduct of their reduction, they were able to show that under the Exponential Time Hypothesis (ETH) of Impagliazzo, Paturi and Zane [JCSSâ01] the STEINER ORIENTATION problem does not admit an f(k)â
no(k/logk) algorithm for any computable function f. That is, the nO(k) algorithm of Cygan et al. is almost optimal.
In this paper, we give a short and easy proof that the nO(k) algorithm of Cygan et al. is asymptotically optimal, even if the input graph has genus 1. Formally, we show that the STEINER ORIENTATION problem is W[1]-hard parameterized by the number k of terminal pairs, and, under ETH, cannot be solved in f(k)â
no(k) time for any function f even if the underlying undirected graph has genus 1. We give a reduction from the GRID TILING problem which has turned out to be very useful in proving W[1]-hardness of several problems on planar graphs. As a result of our work, the main remaining open question is whether STEINER ORIENTATION admits the âsquare-root phenomenonâ on planar graphs (graphs with genus 0): can one obtain an algorithm running in time f(k)â
nO(kâ) for PLANAR STEINER ORIENTATION, or does the lower bound of f(k)â
no(k) also translate to planar graphs
Efficient Enumerations for Minimal Multicuts and Multiway Cuts
Let be an undirected graph and let be a
set of terminal pairs. A node/edge multicut is a subset of vertices/edges of
whose removal destroys all the paths between every terminal pair in .
The problem of computing a {\em minimum} node/edge multicut is NP-hard and
extensively studied from several viewpoints. In this paper, we study the
problem of enumerating all {\em minimal} node multicuts. We give an incremental
polynomial delay enumeration algorithm for minimal node multicuts, which
extends an enumeration algorithm due to Khachiyan et al. (Algorithmica, 2008)
for minimal edge multicuts. Important special cases of node/edge multicuts are
node/edge {\em multiway cuts}, where the set of terminal pairs contains every
pair of vertices in some subset , that is, . We
improve the running time bound for this special case: We devise a polynomial
delay and exponential space enumeration algorithm for minimal node multiway
cuts and a polynomial delay and space enumeration algorithm for minimal edge
multiway cuts
A Tight Algorithm for Strongly Connected Steiner Subgraph On Two Terminals With Demands
Given an edge-weighted directed graph on vertices and a set
of terminals, the objective of the \scss
(-SCSS) problem is to find an edge set of minimum weight such
that contains an path for each . In this paper, we investigate the computational complexity of a variant of
-SCSS where we have demands for the number of paths between each terminal
pair. Formally, the \sharinggeneral problem is defined as follows: given an
edge-weighted directed graph with weight function , two terminal vertices , and integers
; the objective is to find a set of paths from and paths from
such that is minimized,
where . For each , we show the following: The \sharing problem
can be solved in time. A matching lower bound for our algorithm: the
\sharing problem does not have an algorithm for any
computable function , unless the Exponential Time Hypothesis (ETH) fails.
Our algorithm for \sharing relies on a structural result regarding an optimal
solution followed by using the idea of a "token game" similar to that of
Feldman and Ruhl. We show with an example that the structural result does not
hold for the \sharinggeneral problem if . Therefore
\sharing is the most general problem one can attempt to solve with our
techniques.Comment: To appear in Algorithmica. An extended abstract appeared in IPEC '1
Algorithms for Cut Problems on Trees
We study the {\sc multicut on trees} and the {\sc generalized multiway Cut on
trees} problems. For the {\sc multicut on trees} problem, we present a
parameterized algorithm that runs in time , where is the positive root of the polynomial
. This improves the current-best algorithm of Chen et al. that runs
in time . For the {\sc generalized multiway cut on trees}
problem, we show that this problem is solvable in polynomial time if the number
of terminal sets is fixed; this answers an open question posed in a recent
paper by Liu and Zhang. By reducing the {\sc generalized multiway cut on trees}
problem to the {\sc multicut on trees} problem, our results give a
parameterized algorithm that solves the {\sc generalized multiway cut on trees}
problem in time , where time